ARITHMETIC OF LOGIC*

BY E. T. BELL

1. Introduction. This is probably the first attempt to construct an arith- metic for an algebra of the non-numerical genus.f In his classic treatise, An Investigation of the Laws of Thought,J Boole developed the thesis that "Logic (is) • • • a system of processes carried on by the aid of symbols having a definite interpretation, and subject to laws founded on that interpretation alone. But at the same time they exhibit those laws as identical in form with the laws of the general symbols of Algebra, with this simple addition, viz., that the symbols of Logic are further subject to a special law, to which the symbols of quantity as such, are not subject." The special law is what Boole calls the law of duality, x(l—x)=0, or the excluded middle; here the indi- cated multiplication is logical, 1—a: is the supplement of x. Boole showed therefore that abstractly logic is contained in common algebra. Taking rational arithmetic Si ( = the theory of numbers in reference to the positive rational integers 0, 1, 2, 3, • • • , only) and certain parts of the theory of algebraic numbers, particularly the rudiments of Dedekind's theory of ideals and those of Kronecker's modular systems as our guides, we shall see to what extent Boole's algebra of logic may be arithmetized in a precise sense to be defined presently. Although it will be unnecessary to refer explicitly anywhere to the theory of algebraic numbers, it may be mentioned that this theory, which includes rational arithmetic, is a surer guide than the latter in problems of arithmetization. In rational arithmetic the essential abstract structure of the concepts to be extended beyond 0, 1, 2, • • • is often quite ingeniously concealed. This is true, for example, of the G.C.D., L.C.M., and residuation. The theory of ideals, on the other hand, often indicates immediately what transformations by formal equivalence must first be applied to operations or relations of rational arithmetic in order that they shall be significant for sets of elements for which order relations are either irrelevant or meaningless.

* Presented to the Society, San Francisco Section, April 2, 1927; received by the editors Novem- ber 29, 1926. t For the meaning of this term, cf. Whitehead, Universal Algebra, p. 29. X London and Cambridge, 1854; reprinted, Chicago, Open Court, 1916, as vol. 2 of Boole's CollectedLogical Works.

597

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We shall use a few of the commonest notations of algebraic logic ; thus PdQ for P implies Q; P: = :Q for P, Q, are formally equivalent, viz., Pz>Q. QdP, the dot in the last being the logical and; = signifies definitional identity, except where it occurs in conjunction with mod, when it indicates congruence, as in a=ß mod n; the special notation a\ß, borrowed from the theory of numbers, where a, ß are classes, signifies a contains ß, = each element of ß is in a. To avoid confusion we shall never use "contains" in relation to arith- metic; conflicting conventions in this respect have already introduced exasperating paradoxes of language into the theory of numbers. Small Greek letters a, ß, 7, • • • will always denote classes ; S is the set of classes discussed; the null class is denoted by a, the universal class by e, so that w, e are the zero, unity of the algebra of logic 8. Elements of © will be called elements of 8. Logical (but not arithmetical) addition, multiplication of classes a, ß are indicated as usual by a+ß, aß, and if a is any element of 8, the supplement of a is indicated by an accent, a' (instead of the customary bar which is awkward in monotype). Hence a' is the unique* solution of a-f-a' = e, aa' = w, where a is any given element of 8. We shall assume a given set 6 of classes and we postulate that if a, ß are any elements of (£, then a', a+ß, aß are in (5, and, as before, we refer to elements of S as ele- ments of 8. The letters s, p, l, g denote specific operations upon elements of 8, and they are such that (once for all, without further reference) atß for each of t = s,P, l, g is a uniquely determined element of 8 ; the letters c, d, r denote specific relations such that a t ß for each t = c, d, r is uniquely significant in 8. To assist the memory we remark that s, p, l, g, c, d, r may be read sum, product, least, greatest, congruent, divides, residual, terms to be defined in the arithmetic of logic as opposed to the algebra. The least, greatest here are intended merely to recall classes having with respect to given classes properties abstractly identical with those of the G.C.D., L.C.M. with respect to division in 21; these classes g, I are not necessarily the "most" or the "least" inclusive—the rôles with respect to inclusion may be reversed, and g may be either the most or the least inclusive common class of a set, and similarly for I. Hence we shall not use here the names already familiar in Moore's general analysis. The zero, unity in the arithmetic of classes will always be denoted by f, v. It will avoid possible confusion if we add that (f, v) are not necessarily equal to (a, e) respectively.

* For proof of unicity cf. Whitehead, Universal Algebra, p. 36.

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Suppose all the elements, operations and signs of relations, other than the logical constants, are replaced in a set SEof propositions by marks without significance beyond that implied by the assertion of the propositions (which include the postulates of the set). Call the result, C(t), the content of SE. If in C(SE) it be possible to assign interpretations to the marks, giving SE,-, such that SE,-is self-consistent and uniquely significant in terms of the in- terpretation, we shall call SE,-an instance of C(SE). Sets X, (j = l, 2, ■ ■ • ) of propositions having the same content will be called abstractly identical. Our object is to find parts 21,of rational arithmetic 21abstractly identical with parts ?,- of the algebra of classes, hence also with the algebra of relations, and finally with 8. In such a project the following type of invariance under formal equivalence is extremely useful. Let P,- (j = l, 2, ■ ■ • ) be propositions of SEsuch that Pi: ■ :P2. Then the truth-values of Pi oPz, Pz sPi are identical with those of P2 3 Pz, Pz ^Ps- Thus : = : in propositions is abstractly identical with = in common algebra, and in implications a particular proposition may be replaced by any other which is formally equivalent to it. We shall meet several instances of such transformations which are considerably less obvious than the logic which justifies them. The identical transformation of a set of transformations by formal equivalence is that which replaces each proposition of a given set by itself; any set of transformed propositions (including the set transformed by the identical transformation) is called a transform of the original set. Let SE' be a transform of SE,and 21/ a transform of a part 21,-of 21. Then if SE', 21/ are abstractly identical, we shall say that SE is arithmetized with respect to 21,, or simply arithmetized. This kind of arithmetization can be carried much farther for 8 than is done here, but what is given will suffice to show its nature, and it will be evident that at nearly every stage there are alternative ways of proceeding. We shall exhibit arithmetizations of 8 with respect to congruences, the L.C.M., G.C.D., divisibility, primes, and the unique factorization law. Rational arithmetic 21 presupposes the existence of a special ring. In the whole discussion we shall ignore negative numbers, without loss of generality, as the arithmetic (properties of integers as such) which refers to these can always be thrown back to relations between positive integers only, e.g., as done by Kronecker. An abstract ring 3Í is a set © of elements x, y, z, ■ ■ • , u', z', ■ ■ • , and two operations S, P( = addition, multiplication) which may be performed upon any two equal or distinct elements x, y of 3î, in this order, to produce uniquely determined elements xSy, xPy such that the postulates 9Î,-(j=1,2, 3) are satisfied. Elements of © will be called elements of 9Î.

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9îi. If x, y are any two elements of 3Î, xSy, xPy are uniquely determined elements of 9î, and ySx = xSy, yPx = xPy. 9Î2. If x, y, z are any three elements of 5R, (xSy)Sz = xS(ySz), (xPy)Pz = xP(yPz), xP(ySz) = (xPy)S(xPz). 5R3. There exist in 3Î two distinct* unique elements, denoted by «', z', and called the unity, zero of 9Î, such that if x is any element of 9Î, xSz' = x, xPu' = x. These may be compared with the first three postulates of Dicksont for a field, of which they are a transcription, except that the unicity of «', z' is here a postulate, not a theorem, also with Wedderburn'sJ for algebraic fields. In each comparison the omissions are to be particularly noticed. Thus in 9Î we can not infer x = y from xPz = yPz (this inference is in general false for the special rings 3Î' considered later), nor does P have a unique inverse, although we shall later define division. The inference x = y from xSz = ySz also is illegitimate. No attempt has been made in defining 9Î to achieve conformity with other definitions of rings; we are concerned only with isolating from fields what is useful for our project. If S, P and the elements of dt are specialized by interpretation or by the adjunction to 9Î, (j = 1, 2, 3) of further postulates consistent with those for 9Î, or by both of these restrictions, we shall call the result, 9î', a special ring. An instance of 9? is 21. 2. Algebraic congruence in 8. Let xCy be a relation in 9î such that, if x, y, z, w are any elements of 9Î, xCy is uniquely significant in 9î and the postulates (1.1) —(1.4) are satisfied: (1.1) xCy d yCx. (1.2) xCy . yCz: oixCz. (1.3) xCy . zCw:o:(xSz)C(ySw). (1.4) xCy . zCw:z>:(xPz)C(yPw). Then C is called abstract algebraic congruence.

* For the theorem, required later, that f, v are distinct, cf. Principia Mathematica, 1st edition, p. 231, *24.1. It will not be necessary hereafter to prove that ?,- is an instance of $R, as fy*u is the only proposition not immediately obvious from the definitions. t Algebras and their Arithmetics, p. 201. Î Annals of Mathematics, (2), vol. 24 (1923),pp. 237-264,especially p. 240.

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If SRis replaced by its instance 21,an instance of xCy is aCb = (a^b mod m), where a, b are integers = 0 and m is an integer > 0. In 2Í we shall say that C is arithmetic congruence if to the instances in 21 of (1.1)—(1.4) be adjoined the three further postulates (1.5) (a = 0 mod m) : = \m divides a,m 9e 0 ; (1.6) (ka = kb mod m) o (a = b mod?»'), m 9¿ 0, where qm' = m, and q = the G.CD. oí k, m; (1.7) a = a mod m. Any set of propositions in g abstractly identical with any transform of (1.1)—(1.7) will, if true, be said to define arithmetic congruence c in ?. As a practical detail we assign by convention the truth value (+) ( = true) to an asserted proposition, as for example any instance of (1.1), unless it be expressly noted that the value is ( —) (^false). This merely avoids the repeated assertion that our propositions as stated are (+), which they are. We shall now proceed to the partial determination of c by solving (1.1)—(1.4)in 2; the discussion in 8 of (1.5), (1.6) must be deferred until after that of the G.C.D. and the residual in 8. By (1.1) C is symmetric. The only symmetric functions of two classes a, d are (by the laws of tautology and absorption) (2.1) aß, a + ß and their supplements (2.2) a' + ß',a'ß'; (2.3) tx'ß + aß' and its supplement (2.4) aß + a'ß'.

The function (2.3) is that which Daniell* has denoted by \a—ß\ and called the modular difference of a, ß. This is the naturally suggested function for the solution of our problem. It is interesting therefore that it should be rejected by the mildest of the postulates on C, as may be verified in the same way as done presently for another rejection. Now, by evident analogies between 2 and the theory of division for Dede- kind ideals, also by the concept of congruence with respect to an ideal modulus, and further by the "contains" of Kronecker's modular theory, it is immediately suggested that we introduce an arbitrary class p, constant in (1.1)—(1.4), and seek inclusion relations between p and each of the sym- metric functions a in (2.1)—(2.4) to satisfy (1.1)—(1.4).

* Bulletin of the American Mathematical Society, vol. 23 (1916), pp. 446-450.

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The only inclusion relations for two classes 7, 8 are -yjô, y ¿¿a if óV«, and 8\y, y^eii 8^e. We therefore test for each a the truth of the propo- sitions a\u, u\a, either of which may turn out to be (+) or ( —). It will be sufficient to attend to (2.1) for brevity. Hence we are to test these propo- sitions for (3.1)-(3.4), the truth values being unknown, (3.1) aCß = p\aß, p j¿ u if aß ^ u , (3.2) aCß = aß\p, p ^ « if aß ^ t, (3.3) aCß=(a + ß) \p, p ?* e if a + ß s¿ e, (3.4) aCß = p I (a + ß), p ^ to if a + ß 5¿ w. The conclusions are summarized in the following table. (1.1) (1.2) (1.3) (1.4) (3.1) (+) (-) (-) (+) (3.2) (+) (+) (+) (+) (3.3) (+) (-) (+) (-) (3.4) (+) (+) (+) (+) It will suffice to verify the row (3.2) and check the falsity of one (—) proposition, say (3.1).(1.3), deferring consideration of the exceptions. From (3.2) we have aCß=aß\p, and (1.1) is (+), as it must be auto- matically since aß is symmetric. For (1.2) in this case we should have (+) for , 1 a/3 I M-ßy\ ß'. =>:«7| M- Now (cf. Whitehead, loc. cit., p. 43, prop. 14) aI ß .y\ 8: D :ay\ ß8 ; hence

aß\p.ßy\ p'.O laßßy \ pp, : 3 :aßy\ p ; but ay\aßy; hence 07 \p. Again, (1.3) requires

aß\ p .yô\ p: D :(a + y)(ß + i)\ p, which is (+), since (a + 7)03 + 8)\aß.(a + y)(ß + 8) \ y6 ; and (1.4) in this case is aß\ p .y8\ p'. D '.aßy8 \ p which obviously is (+).

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Taking the false (3.1). (1.3) we have aCß=p\aß, and (1.3) demands

p\ aß. p\ yô: 3 :p\ (a + y)(ß + Ô) which clearly is (—). Similarly for the rest of the table. Hence we have the alternative solutions in 2 for the problem of algebraic congruence of classes, (4.1) (a = ßmodp): = :aß\p, p * t, (4.2) (a = ßmodp): = : p\(a + ß), p*w, and evidently either solution can be inferred from the other by the Peirce- Schröder dualism in 2, viz., the reciprocity between logical addition and mul- tiplication. The values of p which must be excepted in (4.1), (4.2) wiU be seen later to be abstractly identical with the excepted modulus zero in 21; in each case the arithmetic zero f is barred as a modulus p. Each problem in 8 has a similar two-valued solution. It is economical however to state both duals in each instance in order to decide readily which must be paired from one solution with one from another to yield the required arithmetic applicable to the simultaneous solutions of several postulate systems. 3. The transform of reflexiveness of congruence. Examining the solu- tions (4.1), (4.2) we see that each violates the simplest property of congruence in 21, viz., reflexiveness. For in 21 we have the (+) proposition (1.7) = (5.1),

(5.1) a = amodffi, for all elements a^O of 21 and m>0. But in 21 we have a —a = 0. Hence we may replace (5.1) by (5.2) 0 = 0modi», since in 21, (0 = 0 mod m) : s : (a = a mod m). Hence, comparing (5.2), (1.5), we may replace (5.1) by its transform (5.2), and hence reflexiveness of congruence in 21may be replaced by the propo- sition that the zero, 0, in 21 is divisible by every element of 21, with the possible exception (removed presently) of "0 divides 0." In 21 we either do not define division by zero, in which case dividends with zero divisors are not in our universe of discourse, or we define division by zero, saying that the quotient is wholly indeterminate, and exclude the process. It is impossible to reconcile either procedure with 2, as will be clear when we come to division in 2, so we make a slight compromise which affects

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nothing in 21 but which is necessary in 8. We shall exclude division by zero in 21 except in the one case where the dividend is also zero, and we shall say that in this case the quotient exists but is wholly indeterminate. 4. Division in 8. Consider in the abstract ring SRa relation having the properties (6.1) xDx, (6.2) xDy . yDz:D IxDz, (6.3) xDy . yDx: d :x = y, where xDy is uniquely significant for each xj^z' (the zero in 9î) and y in 9Î, with the exception (cf. § 3) that z'Dz' is significant but indeterminate in 9Î. These are satisfied in 21by taking xDy=x divides y, and they may be (+) in any instance 3Î' of 9î irrespective of whether division yields a unique quotient.* In 8 we shall select a solution which does not give a unique quo- tient but which does lead to a unique factorization theorem—a rather unexpected situation. As before, analogy with the theory of ideals suggests that we take in 8 an inclusive relation for D. We shall consider both of

(7.1) aDß = a\ß, (7.2) aDß^ß\a, as definitions (in different interpretations) of algebraic division in 8. We may read (7.1) as a divides ß, or ß is a multiple of a, is identical with a contains ß; (7.2) is read a divides ß, or ß is a multiple of a, is identical with ß contains a. Thus (7.1) is as in the theory of ideals; (7.2) is closer to 21. 5. The G.C.D., L.C.M. in 8. These afford interesting examples of invariance under formal equivalence. As first defined in 21, the G.C.D. of a, b is the greatest integer which divides both a and b; the L.C.M. is the least integer which both a and b divide, division as always in 21 being arith- metical, viz., all quotients are required to be in 21. Neither of these is im- mediately applicable to 8. But they may be replaced by their transforms in 21, precisely as in the theory of ideals : With every set of elements a, b, ■ ■ • , h of 21 there is associated a unique element m of % such that every element of 21 which divides each element in the set divides also m ; there also is associated a unique element I such that every element of 21which is a multiple of each element of the set is also a multiple of I.

* If for x^z' the relation xDy implies the existence in91 of a unique element w such that y =xPw, we say that the quotient in 9Î is unique

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The propositions, if true, abstractly identical with these in any special ring 5R'will be taken as the definitions of the arithmetic G.C.D. and L.C.M. in 81'. Abstracting these propositions to 9?, and taking D as in (6.1)-(6.3), we consider two operations G, L upon elements of SRsuch that, if x, y are any elements of 9Î, then xGy and xLy are uniquely determined elements of 9Î, and the postulates (8.1)-(9.4) are satisfied: (8.1) xGy = yGx, (8.2) xG(yGz) = (xGy)Gz = xG)Gz, (8.3) (xGy)Dx. (xGy)Dy, (8.4) zDx. zDy: a :zD(xGy), for G ; and for L, (9.1) xLy = yLx, (9.2) xL(yLz) = (xLy)Lz = xLyLz, (9.3) xD(xLy) . yD(xLy), (9.4) xDz . yDz:o:(xLy)Dz, in all Of which x, y, z are any elements of 'Hit. Note the abstract identity of the pairs (8.1), (9.1) and (8.2), (9.2) with 3îi and the first of 9î2 in § 1, and observe the interesting reciprocal symmetry between (8.3), (8.4) and (9.3), (9.4). A solution in 21of (8.1)-(9.4) is evidently aGb= the G.C.D. of the integers a, b^O, and of (9.1)-(9.4), a7a = the L.C.M. of a, b. Moreover (8.1)-(9.4) together with the postulated unicity of G, L define, or uniquely determine, the arithmetic G.C.D., L.C.M. of elements of 21. Hence we shall call the solution in 2 of (8.1)-(9.4) the arithmetic G.C.D. and L.C.M. in 2, and by taking these properties of the G.C.D. and L.C.M. of classes as fundamental, we automatically fix division and integral elements in 2. There are two solutions, according to the choice of D as in either of (7.1), (7.2). The unicity, essential for arithmetic, is obvious in each instance. To indicate that we have now passed from algebra to arithmetic we shall use d, l, g as stated in § 1, instead of D, L, G, and write the definitions (10.1) adß = a divides ß, (10.2) alß m the L.C.M. of a,ß (10.3) agß sb theG.C.D.ofa,ß, (10.4) (f,u) s the (zero, unity) in 2.

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Assuming for the moment the existence and unicity of f, v, we have the following alternative] solutions of the problems of arithmetic divisibility (d), greatest common divisor (g), least common multiple (I) in 8: (11.1) adß = a\ß, agß^a + ß, cdß = aß, (11.2) adß = ß\a, agß = aß, alß = a + ß. It will be of interest to write down the propositions which are the verifi- cations of (8.1)-(9.4). The first column is for (11.1), the second for (11.2): (8.11) a + ß = ß + a, <*ß= ßa, (8.21) a + (ß + y) = (a + ß) + y, a(ßy) - (aß)y,

(8.31) (a + ß)\a.(a + ß)\ß, a\ (aß) . ß| (aß), (8.41) y\a.y\ß:*:y\(a + ß), a\ y . ß \ y. D :aß | y, (9.11) aß = ßa, a + ß = ß + a, (9.21) a(ßy) = («$7, «+(ß + y) = (a + ß)+y,

(9.31) a\(aß).ß\(aß), (a + ß) \ a . (a + ß) | 7, (9.41) a\y.ß\y:o:aß\y, y \ a . y I ß: s '.y \ (a + ß). 6. Addition (s) and multiplication (p) in 8. A fundamental property in 21of the G.C.D. and L.C.M. is that the productoí the G.C.D. and L.C.M. of two elements of 21is equal to the product of the elements. This determines the choice of p and therefore also of s in 8. As before there necessarily are two solutions. Each must (and does) satisfy dtx, $2 of § 1. Writing (12.1) asß = the arithmetic sum of a,ß, (12.2) apß m the arithmetic product of a,ß, we see that d, s, p must be paired as follows to preserve the property of g, I just mentioned: (13.1) adß = a| ß, asß = a + ß, apß = aß ; (13.2) adß = ß\ a, asß = aß, apß = a + ß. For each of the pairs (11.1), (13.1) and (11.2), (13.2) we have (14) (agß)p(alß) = apß.

7. The arithmetic zero f, unity v, in 8. The algebraic zero, unity in 8 are co, t, so that a+w=a, ae = a. In 8 we must have (15) aîf = a, apv = a,

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for each element a of 2. Hence in each of (13.1), (13.2), f, v are uniquely determined, and we have (15.1) asß = a + ß, apß = aß, (T,u) -(«,«), (15.2) asß^aß, apß = a + ß, (f,u) - («,«). As in 21 the unity in 8 divides each element y, (16.1) add s «Id, v = «, vdy ; (16.2) add = d|«, » - «i xidy. Again, in 21 the only element divisible by the zero in 21 is zero, and in 8 we have, in abstract identity* with 21,

(17.1) Ç\y.(yr*x:).(ï = <»)is(-), (17.2) y|f .(y*r).(r-«)is(-). 8. Arithmetic congruence in 8. The proposition in 8 abstractly iden- tical with (1.5) in 21is

(18) (a = f mod p): D :pda, which is (+) provided (cf. (4.1), (4.2) and (16.1), (16.2)) we pair as follows the definitions of congruence and divisibility in 8:

(18.1) adß = a\ß, (a = ßmodp) = p \ (a + ß), (18.2) adß = ß I a, (a = ßmodp) = aß\ p,

which can be stated together as (19) (a = ßmodp) m pd(asß) = pd(agß). Since (18) is (+) for each of (18.1), (18.2) it follows that the later are necessary for arithmetic congruence c in 2. We have already satisfied (1.7) for c; it remains only to discuss (1.6). 9. Residuals, completion of c, extremes. The transformation by formal equivalence of (1.6) in 21 will complete the sequence of properties of arith- metic congruence c in 8 and yield the abstract identity of congruence in 21, 8. The necessary transformation, suggested by the properties of modular systems, is effected by the abstraction of Lasker'sf concept of the residual for such systems. We shall first abstract to 5R.

* For w |u cf. Principia Mathematica, 1st edition, p. 232, *24.13. f Mathematische Annalen, vol. 60 (1905), p. 49.

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Let a, b, l, m for the moment denote elements of 9Ï. Then, if m is uniquely determined by (w'^the unity in 9î), (20) [aD(lPb)\ . [mDl] . {m ^ u'], where / runs through all elements in 9Î, we shall call m the residual of b with respect to a, and we shall write m = bRa. In 8, (20) becomes

(20.1) {ad(\pß)\ . {pd\\ . [p^v] : = : p = ßra , where r replaces R in the instance (20.1) of (20), and X is an arbitrary class. In 21, the residual of k with respect to m is the quotient of m by the G.C.D. of k and m, viz., this residual is m' in (1.6). The proposition in 8 abstractly identical with (1.6) is therefore

(21) (icpa = Kpßmod p) D (a= mod nrp), and this, as may be verified immediately, is implied by (20.1) and p, d, v as in either of the following columns, which recapitulate previous solutions :

(s) Sum : a + ß , aß , = asß, (p) Product: aß , a + ß , = apß, (g) G.C.D. of a, ß: a + ß , aß , m agß, (l) L.C.M. of a,ß: aß ,a + ß,^alß, (c) a = ßiaodp: p\ (a + ß), aß\p , (f) Zero: to , « , = f, (v) Unity: e , to , = v, (d) a divides ß: a\ß , ß\ a , = adß,

the same interpretations for p, d, v necessarily being taken in both of (20.1), (21). Either column is implied by the other and the reciprocity between logical addition and multiplication. That asß=agß, apß = alß in 8, while the corresponding propositions in 21are ( —), is due to the laws of tautology and absorption, but these do not destroy the abstract identity of the arithmetic of logic and rational arithmetic. The identity is in the fundamental propo- sitions, or postulates, stated abstractly as in 9î, from which 21 is developed, and we have shown therefore that certain parts of both rational arithmetic and the arithmetic of logic are instances of one and the same content. The abstract identity will be enhanced when we find a unique factorization law in 8.

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In passing it may be of interest to note the equivalents in 8 of least, greatest in those parts of 21 which we have abstracted. They are as follows. If in a given set of elements of 8 there be a unique element different from the unity in 8 which divides each element of the set, that element is called the lower extreme of the set ; if in a given set of elements of 8 there be a unique element different from the zero in 8 which is divisible by each element of the set, that element is called the upper extreme of the set. In these defi- nitions either type of division in 8 may be taken ; the upper and lower ex- tremes, viz., the classes which these actually are, in either interpretation are inverted in the other. The G.C.D., L.C.M., residual and congruences in 8 can be restated if desired in terms of extremes. If this be done the verbal forms in 8 become the same as those in 21. 10. Unique factorization in 8. In 21 a set of elements (integers SïO) is said to be coprime if the G.C.D. of all members of the set is unity. Similarly in 8 we define a set of elements to be coprime if their G.C.D. is v. In what follows it is assumed that we are operating in either one of the solutions for s, P, g>I, c, Í, x>,d exhibited in § 9; the results hold in either. For clearness let us recall a few properties of the constituents ( = terms) of a Boole development ( = expansion*) which will be needed immediately. It is assumed that the development is in normal form, viz., that in which all terms with zero coefficients have been deleted. Then first, the logical product of any two distinct terms of a development is the logical zero. Otherwise stated, the terms of a Boole development are a set of classes such that any pair of them are mutually exclusive. The logical sum of all the terms is the logical unity. Hence if a, ß denote any two identical (in which case d—a) or distinct terms of a development, a\ßoa=ß. Second, from a given set of classes we can generate by the operations of logical addition, multipli- cation and taking of supplements a set closed under these operations; the closed set consists of all the elements of 8. The development of the logical unity of this set provides us with a set of terms such that the development of any element of 8 as a function (=logical sum) of such terms is unique. This situation is abstractly identical with the unique factorization theorem in 21, as will be shown in a moment. Suppose now that we have obtained the unique development as above described of a given element of 8. Since 8 is closed under the operation of taking the supplement, it follows that the development of any element of 8 has a dual, obtained by taking the supplements of both sides of the original

* Laws of Thought, Chapter V, especially Prop. III.

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development of the supplement of the given element. This gives us the dual unique decomposition in 8 abstractly identical with the first and with the fundamental theorem of arithmetic. In translating these properties of 8 to arithmetic it is more intuitive to fix the attention on the second column in § 9. That is, we shall think of

asß = aß, apß = a + ß, agß = aß, alß m a + ß,

f = «, v =■ co, adß = ß\ a,

although, by the duality in 8, the theorems are valid in either interpretation. Arranging a few of the abstractly identical theorems and definitions of 21, 8 in pairs, we have the following : (22.1) If the G.C.D. of a, b is 1, then a, b are called coprime. (22.2) If the G.C.D. of a, ß is v, then a, ß are called coprime. (23.1) If k divides the product of a and b, and k, a are coprime, then k divides 6. (23.2) {nd(apß)}. (

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1927] ARITHMETIC OF LOGIC 611

If to obtain the elements of 8 we start from a finite set of classes (or relations) we have in the above arithmetic of 8 a finite image of 21. Conversely the theory of inclusion relations in logic is abstractly identical with rational arithmetic. California Institute of Technology, Pasadena, Calif.

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